As already mentioned above, it would be particularly interesting in this area to attach to each “admissible” representation of a surface group a “canonical” surface, satisfying a specific property, invariant under this surface group in the symmetric space of the Lie group, or in another associated space. A first candidate for the defining property of this surface is to consider minimal surfaces, but the existence and uniqueness of minimal surfaces in the symmetric space of the Lie group is only known in some cases [1]}, and it fails in some key cases, for instance quasifuchsian hyperbolic representations.
